Mathematics

Mathematical modelling of infectious diseases is an interdisciplinary area of increasing interest. Tracking and forecasting the full spatio-temporal evolution of an epidemic can help public health officials to plan their emergency response and health care. We present advanced methods of spatial data assimilation to epidemiology, in this case to the ebb and flow of COVID-19 across the landscape of Spain. Data assimilation is a general Bayesian technique for repeatedly and optimally updating an estimate of the current state of a dynamic model. We present a stochastic spatial Susceptible-Exposed-Infectious-Recovered-Dead (S-E-I-R-D) compartmental model to capture the transmission dynamics and the spatial spread of the ongoing COVID-19 outbreak in Spain. In this application the machinery of data assimilation acts to integrate incoming daily incidence data into a fully spatial population model, within a Bayesian framework for the tracking process. For the current outbreak in Spain we use registered data (CCAA-wide daily counts of total COVID-19 cases, recovered, hospitalized, and confirmed dead) from the Instituto de Salud Carlos III (ISCIII) situation reports. Our simulations show good correspondences between the stochastic model and the available sparse empirical data. A comparison between daily incidence data set and our SEIRD model coupled with Bayesian data assimilation highlights the role of a realization conditioned on all prior data and newly arrived data. In general, the SEIRD model with data assimilation gives a better fit than the model without data assimilation for the same time period. Our analyses may shed light more broadly on how the disease spreads in a large geographical area with places where no empirical data is recorded or observed. The analysis presented herein can be applied to a large class of compartmental epidemic models. It is important to remember that the model type is not particularly crucial for data assimilation, the Bayesian framework is the key. Data assimilation neither requires nor presupposes that the model of the infectious disease be in the family of S-I-R compartmental models. The projected number of newly infected and death cases up to August 1, 2020 are estimated and presented.

Provincial and US state case, hospitalization, and death data can be characterized by relatively long periods of nearly constant growth/decline along with some large outbreaks. This talk will compare the spread in the different jurisdictions and how it has changed with relaxed social distancing measures.

This is the prototype of an agent based model for a closed universe of a population experiencing a contagion-based epidemic, in which risk factors, movement, time of incubation and asymptomatic infection are all parameters. The model allows the operator to intervene at any step and change parameters, thus analytically visualizing the effect of policies like more testing, contract tracing, and shelter in place. Under current development, CovidSimMV is an ABM that supports a Multiverse of different environments, in which agents move from one to another according to ticket with stops. Each universe has its own characteristic mix of residents, transients and attached staff, and persons are able to adopt different roles and characteristics in different universes. The fundamental disease characteristics of incubation, asymptomatic infection, confirmed cases will be preserved. The Multiverse model will support a rich diversity of environments and interpersonal dynamics. These are JavaScript programs that can be run in a browser as HTML files. The code is open source, and available on github.com/ecsendmail.

Given a probability preserving system (X, \mu, T) and a set U of positive measure contained in X we denote by N(U,U) the set of integers n such that the measure of U intersected with T^n(U) is positive. These sets are called return-time sets and are of very special nature. For instance, Poincaré recurrence theorem tells us that the set must have bounded gaps while Sarkozy-Furstenberg theorem tells us that it must have a square. The subject of this talk is a very old question (going back to Følner-1954 if not earlier) whether they give rise to the same family of the sets as when we restrict ourselves to compact group rotations. This was answered negatively by Kříž in 1987 and recently it was proved by Griesmer that a return-time set need not contain any translate of a return-time set arising from compact group rotations. In this talk, I will try to sketch some of these proofs and give a flavour of results and questions in this direction.

Diffusion is the enemy of life. This is because diffusion is a ubiquitous feature of molecular motion that is constantly spreading things out, destroying molecular aggregates. However, all living organisms, whether single cell or multicellular have ways to use the reality of molecular diffusion to their advantage. That is, they expend energy to concentrate molecules and then use the fact that molecules move down their concentration gradient to do useful things. In this talk, I will show some of the ways that cells use diffusion to their advantage, to signal, to form structures and aggregates, and to make measurements of length and size of populations. Among the examples I will describe are signalling by nerves, cell polarization, bacterial quorum sensing, and regulation of flagellar molecular motors. In this way, I hope to convince you that living organisms have made diffusion their friend, not their enemy.